## Abstract

We provide further insight into why the inverse rule [J. Opt. Soc. Am. A **13**, 1870 (1996)] for multiplying two finite Fourier series of two pairwise discontinuous functions yields correct results at the point of discontinuity.

© 2000 Optical Society of America

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### Equations (9)

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(1)
$$f(x)=\left\{\begin{array}{c}a,-\mathrm{\Lambda}/2x0\\ b,0x\mathrm{\Lambda}/2\end{array}\right.,$$
(2)
$$g(x)=\left\{\begin{array}{c}b,-\mathrm{\Lambda}/2x0\\ a,0x\mathrm{\Lambda}/2\end{array}\right..$$
(3)
$${h}_{T}^{(2)}(0)=\frac{1}{\left[\left(\frac{1}{a}+\frac{1}{b}\right)/2\right]}\frac{a+b}{2}=\mathit{ab}=h(0),$$
(4)
$$\frac{1}{\left[\left(\frac{1}{a}+\frac{1}{b}\right)/2\right]}.$$
(5)
$${h}_{T}^{(1)}(x)={f}_{T}(x){g}_{T}(x)=[{c}_{0}+{D}_{N}(x)][{c}_{0}-{D}_{N}(x)],$$
(6)
$${D}_{N}(x)=\sum _{n=1}^{N}{c}_{n}sin\mathit{nKx},\hspace{1em}\hspace{1em}{c}_{0}=(b+a)/2,$$
(7)
$${c}_{n}=(2/\pi n)(b-a),\hspace{0.5em}n\hspace{0.5em}\mathrm{odd},\hspace{1em}\hspace{1em}{c}_{n}=0,\hspace{0.5em}n\hspace{0.5em}\mathrm{even},$$
(8)
$${h}_{T}^{(2)}(x)=(1/{f}_{T}^{\mathrm{REC}}){g}_{T}(x)=\frac{1}{(1/\mathit{ab}){g}_{T}}{g}_{T}=\mathit{ab},$$
(9)
$$\mathrm{for}\hspace{0.5em}\mathrm{all}x.$$